If an angle of $\displaystyle 20$ degrees were constructible so would a $\displaystyle 360/20=18$ sided
polygon.

The first three Fermat primes are $\displaystyle 3,\ 5,\ 17$, clearly $\displaystyle 5$ and $\displaystyle 17$
do not divide $\displaystyle 18$, so for the $\displaystyle 18-gon$ to be constructible
$\displaystyle 18$ would have to be a power of $\displaystyle 2$, or $\displaystyle 6$ (as it is $\displaystyle 18/3$) would
have to be a power of $\displaystyle 2$. They are not so the $\displaystyle 18-gon$ is not
constructible and so an angle of 20 degrees is not constructible.